Square-root conjecture for geometric Whittaker complexes

Let $X$ be a smooth projective curve, let $E$ be an $\operatorname{SL}_2$-local system on $X$, and let $\operatorname{RCov}^d$ be the moduli stack of degree-$d$ double coverings of $X$. For a local system $V$ on $X$, let $CL_V^d$ be the complex on $\operatorname{RCov}^d$ whose fibre at $({\cal E},s_2)$ is

$$\bigoplus_{\theta\geq 0}\operatorname{R\Gamma}(X^{(\theta)},(V\otimes{\cal E}_0)^{(\theta)})[\theta].$$

Here ${\cal E}_0$ is the rank-one anti-invariant local system associated with the double covering.

Square-root conjecture. For any $d\geq 0$, there is a complex ${\cal S}_E^d\in\operatorname{D}(\operatorname{RCov}^d)$ equipped with an isomorphism

$$({\cal S}_E^d)^{\otimes 2}\\,\widetilde\to\\,CL_E^d.$$

Moreover, ${\cal S}_E^d[\dim\operatorname{RCov}^d]$ is Verdier self-dual, and the complex sought in the source identifies canonically with ${\cal S}_E^d$.

The conjecture proposes a geometric square root of the central-value complex, motivated by Whittaker models and classical formulas for Whittaker coefficients. The source presents it as conjectural without supplying a resolution.